Early live TV shows were preserved by filming them directly off the screen. The quality could be pretty good if not comparable to shooting on film, but without kinescopes, many things would be gone forever.

Here is the funniest thing ever broadcast.

“Sentimentality is when it doesn't come off—when it does, you get a true expression of life's sorrows.” —Alain-Fournier

One important point to understand is that unlike motion picture cameras, image orthicon cameras used for live TV did not contain a magazine for film (with one exception, discussed below). Up to 1947, televised images generally weren't recorded at all, save for amateur or experimental efforts. A method of preserving TV programming produced with electronic cameras was necessary if nationwide network television was going to happen. (The only alternative would've been to re-stage live broadcasts three hours later for the West Coast, a costly option and obviously impossible for non-scripted shows.) Kinescope films were in effect the only time-shifting method available for live programming from 1947 until Ampex introduced the first practical videotape recorder in 1956.

Kinescope films were inherently a compromise; they preserved an approximation of the live TV picture, but not its exact appearance. Rather than having a "live" appearance, kinescopes carried a "filmed" look, and not one as clean or crisp as a motion picture camera could produce. Nonetheless, by the early 1960s it was possible to produce quite a good-quality kinescope film, as evidenced by "What's My Line?" episodes from that period.

The short-lived DuMont Electronicam tried to combine the best of both worlds, i.e. of motion picture and image orthicon cameras. The image orthicon camera would transmit the live picture, while the motion picture camera would preserve the image as if it were a filmed series or theatrical movie. "The Honeymooners" was produced this way. A kinescope was made of the live broadcast, then the film from the camera was edited to match it. The Electronicam was rendered obsolete soon after the VTR was introduced.

Even after videotape was introduced, kinescopes continued to be used for archival purposes and to send programs to small stations that lacked tape machines. Videotape could be erased and reused, and that's exactly what happened to many programs (like, again, "What's My Line?"), some of which were saved only on kinescope if at all. Kinescopes were made of both live and pre-taped programs well into the 1960s, with ABC doing so as late as 1970.

...And the Brits call it a telerecording. The BBC even has a technique to bring back the colour signal from b/w film, because the colours left a pattern of tiny tiny dots.

For most 1950s-1960s shows the filmed copy in fact more or less became the master tape; the tapes were reused. People discussing lost TV programs often get this wrong and refer to wiping and tapes when it's mostly about 16mm film being junked. There are some examples on youtube where they have film AND video tape and compare the two for quality. The picture will be cropped; sound quality suffers too; film - usually - used optical soundtracks which just isn't quite as good.